Spherical tilings by congruent quadrangles: Forbidden cases and substructures

Yohji Akama,Nico Van Cleemput

doi:10.26493/1855-3974.581.a6a

Yohji Akama, Nico Van Cleemput

Open Access

https://doi.org/10.26493/1855-3974.581.a6a

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Abstract

In this article we show the non-existence of a class of spherical tilings by congruent quadrangles. We also prove several forbidden substructures for spherical tilings by congruent quadrangles. These are results that will help to complete of the classification of spherical tilings by congruent quadrangles.

Highlights

In this paper we prove the non-existence of a subclass of spherical tilings by congruent quadrangles which have three equal sides and one side different
We show that there exists no spherical tiling by congruent quadrangles of type 2 if the quadrangles are isosceles
We show that the most symmetric of type 2 quadrangles, i.e., the isosceles quadrangles of type 2, cannot be used to tile the sphere. This might seem surprising, since spherical tilings by congruent quadrangles of type 2 do exist, but it can be explained because being isosceles and tiling the sphere forces the quadrangle to be of type 1

Summary

Introduction

In this paper we prove the non-existence of a subclass of spherical tilings by congruent quadrangles which have three equal sides and one side different. We list several forbidden substructures for this type of spherical tilings It follows from Euler’s formula that spherical tilings by congruent polygons can only exist for triangles, quadrangles and pentagons. Akama and Sakano [7] completed the classification of spherical tilings by congruent kites, darts and rhombi Since these quadrangles can be subdivided into congruent triangles, they could rely on the classification by Ueno and Agaoka to solve this classification. Nakumara and Sakano [1, 2, 7] showed that if concave quadrangles are allowed, there exist several tilings which have non-congruent tiles but for which the inner angles and the underlying graph are the same. We look at the different possible configurations of angles around each vertex and we use this to show some forbidden substructures for the underlying graph

Definitions

Spherical tilings by congruent isosceles quadrangles of type 2

Conclusion